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ICMS/EMS Postgraduate Courses

February - May 2002

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Numerical/Applied Mathematics
Wednesday 8 May 2002, from 1430
Maths Dept, Livingstone Tower, University of Strathclyde,
26 Richmond Street, Glasgow G1

Talks will be held in L8.31 (level 8). Tea will be in L9.20
The link for a map is: http://www.strath.ac.uk/maps/livingstone.htm

1430 - 1600 Roger Fletcher
Nonlinear Systems and Optimization .
1600 - 1630 Tea
1630 - 1750 Andrew Lacey , Heriot Watt
Qualitative Behaviour for Partial Differential Equations.

ABSTRACTS
Roger Fletcher
Nonlinear Systems and Optimization
Most applied mathematicians need to solve nonlinear algebraic systems or optimization (variational) problems at some time in their career. Yet knowledge of modern techniques and how to access them is often sketchy. A summary of the main features of the subject will be presented.
  • Linear systems and systems with finite termination
  • Nonlinear systems and Optimization
  • Newton-Raphson method for systems
  • Lagrange multipliers and their significance
  • Sequential Quadratic Programming (SQP)
  • Very large systems
  • Discrete variables
  • Optimization modelling languages


Andrew Lacey
Qualitative Behaviour for Partial Differential Equations
This lecture will look at a variety of qualitative features exhibited by different types of PDE. These include shock formation by solutions of nonlinear first-order hyperbolic equations and smoothing properties of parabolic problems. The role of specific terms in some equations, such as the promotion of 'blow-up' by some reaction terms, will also be highlighted. The consequences for certain models, e.g. simple models for traffic flow or heat equations, will be examined.

Geometry and Topology
Wednesday17 April 2002, from 1430 ICMS, 14 India Street, Edinburgh EH3 6EZ

1430 - 1515 Elmer Rees, Edinburgh
Introduction to Symplectic Geometry.
1515 - 1545 Tea
1545 - 1630 Bernd Schroers, Heriot Watt
The role of symmetry in classical and quantum mechanics.
1640 - 1725 Sergei Merkulov, Glasgow
Deformation quantisation and Kontsevich's formality theorem.

ABSTRACTS
The theme of the lectures will be Symplectic Geometry. It is an old topic with its roots in classical mechanics but in recent years it has become increasingly useful in several other areas of mathematics. Topics in which it is used include partial differential equations, spectra of matrices and quantum mechanics.

Elmer Rees, Edinburgh
Introduction to Symplectic Geometry .
Manifolds and their cotangent bundles. Lie groups, left invariant forms. Symplectic structures. Symplectic quotients, moment maps and convexity. Application to the study of the spectra of n X n Hermitian matrices (including Atiyah's theorem).

Bernd Schroers, Heriot Watt
The role of symmetry in classical and quantum mechanics.
Symmetries are of great practical importance in both classical and quantum mechanical problems. They also provide structural guidance in the transition from classical to quantum mechanics. In the lecture I will illustrate these statements with a simple but surprisingly rich example: the Kepler problem (motion of a massive particle in a 1/r-potential). Topics to be covered: symmetries and conservation laws, dynamical symmetries, reduction and enlargement of phase space.

Sergei Merkulov, Glasgow
Deformation quantisation and Kontsevich's formality theorem.
We discuss some of the basic ideas behind Kontsevich's celebrated formality theorem and discuss its application to the deformation quantisation on Poisson manifolds.


Applied Analysis/Dynamical Systems
Wednesday 20 March 2002, from 1430 ICMS, 14 India Street, Edinburgh EH3 6EZ

1430 - 1545  Periodic Solutions of Scalar Equations - Jack Carr  
1545 - 1615  Tea  
1615 - 1730  Circle Maps - Sandy Davie  

ABSTRACTS
Periodic Solutions of Scalar Equations (Jack Carr)
The main aim of this lecture will be to present some of the ideas of dynamical systems in a simple setting. The problem we will study is the scalar ordinary differential equation x'(t) = g(x) + f(t)
with f(t) a periodic function. Some of the concepts we will explore are Poincaré maps, bifurcation of periodic solutions, stability, and variational equations.

Circle Maps (Sandy Davie)
This lecture will consider the dynamics of mappings of the circle to itself. Such mappings occur frequently in the study of dynamical systems. Understanding circle maps requires methods of topology, number theory and analysis as well as numerical calculations. The lecture will describe how circle maps can arise in practice and illustrate some of the methods used in their study.


Tools of the Trade: A Practical Guide to Postgraduate Study
Wednesday 27 February 2002, from 1430 ICMS, 14 India Street, Edinburgh EH3 6EZ
1430-1500 How to get a PhD - Jack Carr - Click here for the abstract
1500-1530 Elements of report/paper writing - Penny Davies
1530-1600 Latex for reports and presentations - David Griffiths
Click here for the PostScript and LaTeX source files for the talk.
1600-1630 Tea
1630-1700 Mathematical/bibliographical software - Dugald Duncan
1700-1715 What happens in a PhD oral - Alastair Gillespie
1715-1745 What industry wants from mathematicians - Paul Moseley
1745-1900 Informal discussion and drinks
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